Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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138
From
(3.25)
we have
Chapter
3
-b
Let
w
be an
arbitrary
element of
L2(0,T;
vl).
Then, from the
preceding formula we get
-
f(
@(w),u-
w)dt=
lim
Let us set here%
=
?I
+
A
5,
for any
A
>
0
T
-8++
T+
-b-8
+
0
E+O
IT-0,
0
lim
f
(
p(u~)+(w),u~-
w)dt
2
0.
-P
where
A
>
0,
(1,
E
L~(o,T;
v,).
Then,
T
-t
-+
+I(@(u+A+)dt-
20.
0
Going over to the
limit
at
A
-t
Owe conclude thaJ
@fi)
=
0,
-f
i.e.
u
=
Pk
;*Therefore,
if
in
(3.27)
we assume
Q(t)
EK
and
go
over to the
limit
at
E
+
0
then we deduce that for the limiting
functions
(
c,
lineq quality
(3.14)
is
valid i.e.
(
z,
p)is
a
generalized solution of problem (3.1)
-
(3.3).
Thus, Theorem 3.1
is
proven.
6.
Other formulations
of
one-side problems
Let us now formulate several variants of the problem
of
flowing
through with one-side conditions, whose solvability
is
proved in
an analogous way a3 in
(3.1)
-
(3.3).
Formulation
2.
On the portion of in-flowing we set the total
vector velocity and density:
++
(3.29)
1 1
++
u
=
u
,
p
=
p
,
(x,t>
E
s',
(
ul,
n
<
o
.
On
and the normal velocity component be non-negative:
I'
we set the tangential components of the velocity be known
(
3.
?.I=
11
g.
,
i=
1,2;
p
=
1;
(3.
3)
20,
(x,
t)
E
S'
.
(3.30)
On
the
solid
w~ll
Po
the no-slip condition holds
(3.31)
-b
u
=
0
,
(x,
t)
E
so
.
0
Formulation
3.
On
the parts
$
(3.31) are given, while on
I
,
instead of the pressure in condi-
tion (3.30)) the total stress
and
I"
conditions
(3.29)
and
is
known. Thus, formulation
3
is
characterized by the boundary
conditions
(3.291,
(3.31) and (3.32).
One can also consider the problem with the total stress given both
at the inlet
I#'
and at the outlet
?.This
variation of the
Initial-Boundary
Value
Problems
139
problem
of
flowing through
is
described with the followi.ng condi-
t
ions
.
Formulation
4
--t
a)
U=O
Y(X
b)
(
"u-
ti)
=
0
t)
E
so
,
i=1,2;
p=
p';
p
+
I/
2
p
lu
1'
=
h,
;
-t
(3
-
n
SO,
(x,t>
E
S"
(3.33)
++
i=1,2;
p
+-/'
p
13
1'
=
h
;
(u.
n
)
2
Oy
(x,
t)
E
s2
.
For
simplicity, the boundary data for the tangential velocity com-
ponents are again assumed to be zero.
It
should be also recalled, that of interest
is
also the problem
with the pressure given both on
1."
and
1"
Formulation
5.
In conditions
(3.33)
b) and c) instead of the
stress
p
+
1/2
p131'we know the pressure
P
=
P"(X,t>,
(x,t>
ES';
p
=
P2(XYt),
(XYt)
6
s'
(3.34)
It
should be added, however, that
in
this problem the
first
a priori estimate
in
the energy norm
is
local with respect to time,
since the non-linear convective terms in the pulse momentum equa-
tions
(3.1)
yield non-zero integrals over the boundaries
r2
which are
of
cubed order with respect to velocity.
In
more detail, when multiplying
(3.1)
by
?i
of the type
r'
and
there arises a relation
wherefrom we can get the estimate
only at a sufficiently
small
value of the quantity
Therefore,
solvability of the problem in such a formulation even
in the class
of
generalized solutions
is
quaranteed only
'Iin
the
ma1
1".
4.
A
MODEL
OF
AN
INHOMOGhT~OUS
FLUID
WITH
DIFFUSION
Papers
c49,
651
present us with
a
simplified mathematical model of
a two-component mixture, with a diffusive
mass
exchange
among
the
medium particles
of
various density accounted for. The equations
of the model are as follows:
--t+
140
Chapter
3
+
-,
ap
-,
-b
=
p
A
u
-
V
p
+
pf,
-
+
(u
v)p
=
A
A
p,
div u
=
0,
at
-
(4.1)
+
where u
is
a
mean (volume) velocity,
p
is
the mixture density,
A
=
const
>
0
is
the diffusive coefficient
in
the
Pick
law,
p
=
const
>
0
is
the
viscosity. For
A
=
0
equations (4.1) trans-
fer to model
(1.1).
A
characteristic mathematical peculiarity of
system
(4.1)
lies in the fact that
it
is
non-diagonal in
its
main
part,
which hampers the investigation
of
the boundary problems for
system (4.1)
as
compared
with
(1.1).
Let the motion
of
the mixture take place in
a
bounded domain
Q
with
the boundary
r
c
Cz.
Let us assume that
I'
is
impermiable
and there
is
no mass exchange between the mixture and the
external
medium through the boundary
r.
This
requirement can
be
formulated
in the
form of the boundary conditions
--*
aP
ulr
=
0,
-1
=o.
(4.2)
an
r
--*
Here
n
is
a
unit external normal to
I'.
Besides,
it
is
necessary
to set the initial conditions
(4.3)
We can consider another boundary condition on
I?
valup of density:
Aa
far
as
the initial density
po(x)
in (4.4) that they are positive and founded functions,
o
<
m
I
po(x>
5
&I
<
m,
m
5
p(x,
ti>
I
M.
when
we
know the
(4.4)
"Ir
=
0,
plr
=
p
(x,
t),
(x,
t)
E
S
=I'
x
(0,
T).
as
well
as
p'(x,
t)
we
set
(4.5)
1
Let
us
now consider problem (4.11, (4.21,
(4.31,
introducing the
necessary notions and notations.
As
in
$4
1,
we
shall
need the
spaces
J(Q)
and
J'(Q)
of the solenoidal ve2tor-functions,
as
well
as
the subspace
W
of
the scalar space
W,(Q)
of
the functions
+(XI
lized solution
of
problem (4.11, (4.21,
(4.3).
For this purpose
let
us
first
show
a
divergent presentation of the momentum equa-
tions in system
(4.1):
which consists
wherein a(,/anlr =O.Let
us
define
a
genera-
aP
-
a
4
-
(pui)
+
div(puiu
-
Aui~
p
-
A
-
u)
=
p
A
ui
-
at
axi
ap
axi
-
--
+
pfi,
i
=
1,
2,
3.
-#
Definition 4.1. The functions
(u,
p>
are called
a
generalized
so-
lution of problem (4.11, (4.2). (4.31,
if
Initial-Boundary
Value
Problems
141
b)
diffusion
equat&,on in system
44.1)
12 satisfied
a.e. in
c)
for every
I.L
CP
€
C
(0,T;
J
(Q)),
Q(T
)
=
0
,
the next in-
y
tegral identity
I
*
+
-+
++
1
{(P
$,at
+
(U
-V)Q
)2,B-
p(
u
,Q
)
-
0
J'(L-2)
is fulfiled.
Let,
us
formulate the main results.
Theorem
4.1.
If
uo
E
J(Q>,
p
E
[1,2],
q€
[6/5,
21,
l/p
+
3/
2q5
7/4,
and
constants
-i
-B
Po
E
Wi(Q),
fELp(O,TC; Lq(Q)),
A,
p,
IC
and
m
are such that
A
<
2
p
(iii
-
lil
I-''
(4.7)
then there exists at the least one generalized solution
of
problem
(4.11, (4.2),
(4.3).
Theorem
4.2.
If
additionally
problem has a unique strong solution locally in time. This solution
is global in the two-dimensional plane case.
+
-D
Uo
E
J'(Q>,
f
E
Lz(
'.i
),then the
142
Chapter
3
The proof
of
the existence theorem is the same as in the case
h
=O.
We shall give only the
a
priori estimates. Firstly, the parabolic
equation for the density
by
the maximum principle gives
0
<
1il5
p(x, t)
Ski
<
m
(4.9)
Multiplying
(4.8)
by
P(X,~)
and integrating over
4
leads to
est imate
(4.10)
The impulse equation
(4.1)
gives the energy relation
In the first Lerm, after integrating twice by part, we oStain
Then
the equality
gives
-3 -3
bl+m
3
au,
au
((U*V)VP,U)
=J(p--)
c
-
--;idx
28Q
Q
2
i,j=l
axj
a5
Initial-Boundarv
Value Problems
143
1
1
2
2
From the inequality
Ip-
-
(M+m)(5
-
(&I-m)
we conclude
h
2
By the condition
(4.7)
p-
-
(M-m)=
p,
>
0
and
so
As in
8
1
we have
(4.11)
From the equation
(4.81,
after multiplying by
Ap(x,t)
it is
easy
to
get the estimate
And, as in the lemma 1.2, we get the continuity with respect
to
time
in
~,(q)
-
norm
The estimates
(4.9)-(4.13)
give the existence theorem
4.1
in the
class of generalized solutions.
Theorem
4.2 is proved as lemma 2.2.
5
5
Model of a medium with intrinsic degrees of freedom.
Let
us
consider, the system of equations
(1.44)
from ch.
1
describing
144
Chapter
3
the motion
of
a granular medium which for simplicity
is
assumed ho-
mogeneous,
p
=
const
>
0:
3
au
-9
3 3
3-b
+
-
+(u*V)u+Vp=f+
[Wx
u]+VAU,
at
aw
3
-P
+3 3
-
+
(
U*
0)
W+
F
(p)
3
(5.1)
W=
‘p,
div
U
=
0,
(
X,
t)
E
Q=
Q
x
(0,
T)
at
Xere
FZ(p)
is the given non-negative function from
C‘(0,
a)
and
We shall consider the next boundary and initial conditions
?he pressure
p
must be non-negative:
iuin
p(x,
t)
=
po
(t)
20
.
xeSL
Without
loss
of
generality we
can
put
po
(t)
p
0
:
min
P(X,
t)
=
o
,
t
E
[o,
T
]
.
Ye5
(5.3)
(5.4)
(5.5)
+
The equations for are the system of the first order. Any of these
equations
is
analogous
to
the equation for density pin the modrl
of
inhomogeneous fluid.And the single distinction is associated with con-
Initial-Boundary Value Problems
145
dition on the pressure (5.5). Let
us
now consider the scheme
of
proving the theorem of existence. Having
set
the vector
in an arbitrary way, let us substitute
it
into the pulse equations
-3
w
=
3,
-3
au
-+
-9
-
+
(u,
v)u,+
p,=
f
+
q[
w1
x
u,]+
vA
5,)
div
u,=
0.
(5.6)
at
-
Herefrom, using
the
boundary and initial
conditions, we can deter-
mine
5,
and
v
p,
in
which case
if
W
,
and
3'
are sufficiently
smooth functions, then
the
solution
GI,
v
p,)
will
be
also
smooth.
(in
the
two-dimensional problem
in
the
whole in time,
Fn
the three-
-dimensional one
-
generally speaking, only
in
the
small).
Then
then a new vector
of
the angular velocity
3
=
G2
the linear equations
p,(x,t)
is
determined
by
~~,(~,t)
and
the
condition (5.51, and
is
found from
The coefficients
of
this
system contain the obtained functions
and, thus,
there
arises
the mapping
:
2,
--t
J2
the
fixed po-
ints
of
which, together with the corresponding
d,
and
P,
give the
solution to problem (5.1)
-
(5.5).
Let
us
now describe the derivation
of
the a priori estimates.
First,
from the momentum equations, through multiplying
by
in
a
scalar way in
L2(
a)
we get
In
this
case
we
allow for the fact that
the
mixed product
(L".
x
31-
deduce
:
+
equals zero. Then from the equations for
2
we
1;
lo,k<
c
*
(5.9)
Returning now to the pulse equations, we get a stronger estimate
for
the
velocity and pressure:
(5.10)
-i
11
ll\v2,,
+
11
VP
llLpJ
=
c
+J
2.
(.%)
1
In
this
case
(5.90)
1s
a global estimate for a two-dimentional
flow and a local one for
a
three-dimentional flow.
Differentiating
the
equations for;
with
respecto to
x,
multi-
plying by
zx
and integrating with respect to
Q
,
as
well
as
allowing for (5.101, we deduce:
71
146
Chapter
3
From conditions (5.2) for
Q
3
we have by (5.5)
zm
IP
lo&
5
c
II
VP
1Iq,Q"
1lP.F'
VP
IIq,Q
5
c
II
VP
II,,
Q
'
Hence,
at
Y>
max
(
3,
2m
(5.
-9
The equations for
W
give a direct estimate for the derivative
with
respect to
t
:
(5.
According to the theorem
of
imbedding,
it
follows from (5.111,
(5.12) that at
9
>
4 the function
w
respect to
x
and
t
:
-B
is
continuous
by
Hzlder
with
(5.13)
lz
I,,,
SC,
o!
=
(q
-
4)
/
q.
Then, from the momentum equations we again conclude:
I
u'
I2ia,1+
Cr/q,dl
+
10
p
IcL,a/2,(4:
*
(5.14)
These properties
of
smoothness make
it
possible to unambiguously
restore
P(x,t>
as
a
continuous in
p
function by the gradient
data which are required for obtaining the estimates (5.8)
-
(5.14)
and, hence, for proving the theorem
of
existence.
Theorem
5.1.
Let the domainn be bounded, simply-connected, with
the boundary
Let
2
E
e901/z
(%I,
"u0c
c2"(Q
),
6
and condition (5.5). Let us determine the Conditions for the
of
class
Cz+
-#
cp
E
L~(o,Y;
~~(2))
n
L,(o,T;
sic
a>>,
wo
E
:i'(
Q>,
9
o
<
CL
<
I,
q
>-
rnax(4
/
(1-
a>,
2m
1,
and the necessary conditions for
fitting
the initial and boundary
data be met. Then problem (5.1)
-
(5.5)
has
a unique solution
with
properties (5.8)
-
(5.14) within a sufficiently
mall
length
of
time
LO,
Tj
In the two-dimensional case the solution
is
extended
to any finite time interval.
Proof. The solvability
is
proved in uite a conventional way on
the basis of estimates (5.8)
-
(5.147. We shall elucidate only the
problem of uni ueness, since there
is
an additional non-linear
condition (5.53. Forming
a
difference between the two possible
so-
lutions
(
ul,
p',
;'>
and
(
u2,
Pz
9
Efz
):
3
=
;1-;2,
p
=
pz,
tf
=
$1
-
-+z
w
we get for
(
2,
p,ij>
a non-linear system
of
equations
Initial-Boundary Value Problems
147
++
A
=
w‘[
Fz(
PI)- Fz(pz)
](
pl-
p2)-’
I
As
far as P
(5.51,
their difference
P=
PI-
Pz
one point
8:
E 5
at
any fixed
t
and, hence,
From the
first
equation
(5.15)
we have
and P2 are non-negative functions with the property
must equal zero
at
least at
lP(t)
lo,$
c
IIVP
II,,Q
9
¶>
3
+
II
v
P
II
5
c
11;;
(dt
=
Qx (0,t).
11
Il;12.1(%)-
9
s*l;t
From the second one, in
its
turn, we get
m
+
From these two relations
it
is
evident that
u
=
0,
=
0,
P
=
0.
6.
PROBLEMS UNSOLVED
1.
In the theory of the Navier-Stokes equations the problem
of
the
uniqueness
of
generalized solutions
is
the most difficult one. For
sufficiently smooth solutions, the uniqueness
is
proved quite
easily, while in the class of generalized functions the problem of
uniqueness (or non-uniqueness) remains open to discussion. Even in
the case of
a
homogenous fluid
(P
‘const>
0)
in
the three-di-
mensional problem,
it
has not yet been established whether the ge-
neralized Hopf solution
is
unique,
or there may appear new solu-
tion with growing time.
In
the two-dimensional problem the theorem
of uniqueness
is
valid
L
88,
961.
2.
Not less difficult
is
the problem on
a
limiting transition with
respect to viscosity, when the coefficient tends to zero. The
problem has been
a0
far
studied only in
a
linear formulation
[
881,
while for non-linear equations, the situation when there are no
olid boundaries of the domain of
flow,
has been investigated (see
f38,
SSJ
1.
3.
Of
great interest for the mathematicians are the problems with
da ies. The
first
results here were obtained by V.V.Pukh-
~~~~e~~12f and
V.A.
Solonnikov
[
126,
1281. Note, however, that
their investigations refer to the case
of
a homogenous fluid.
As
to the equations of
an
inhomogenous fluid, there are
still
no re-
sults, concerning the solvability of problems with open boundaries.
4.
No problems concerning the correctness
of
stationary problems
for the equations of an inhomogenous fluid have been studied. The
existence of generalized solutions can,
in
all likelihood, be
proved with the method of
E
-regulation, through adding an ellip-
tical operator with a small parameter into the equation for the
density
p.
The uniqueness of stationary solutions has not yet
been proved for great Reynolds numbers.